Single neuron computation
(חזרה אל פיזיולוגיה ב) Ramon Y Cajal postulated “The Neuron Doctrine” (Noble 1906): 1. Neurons are independent 2. Information flows (from spines to dendrites to axons) Neuronal Facts A little bit of statistics in cortical neurons: Morphology Diameter near soma: 1 - 6 µm Tip diameter: 0.3 - 1 µm Average path length: 0.15 - 1.5 mm Total dend. length: 1 - 10 mm Dendrite area: 2,000 - 750,000 µm2 Dendrtic trees/neuron: 1 - 16 Dendritic tips/neuron: 10 - 400 Dend. spines/neuron: 300 - 200,000 Spines density/1µm dendrite: 0.5 - 14 Spine length: 0.1 - 2 µm Spine neck diameter: 0.04 - 0.5 µm Spine head diameter: 0.3 - 1 µm Spine volume: 0.005 - 0.3 µm3 Physiology Passive properties* Membrane. Resistivity, Rm: 5 -100K Ωcm2 Axial resistivity, Ri: 70 - 250 Ωcm Membrane. Capacitance, Cm: 1 - 2µF/cm2 Membrane time constant, τm: 1 - 100 msec Dendrite space constant, λ: 0.2 - 1 mm Elecrotonic length, L = x/λ: 0.2 - 2 Soma input resistance, RN: 1 - 103 MΩ Input resistance at terminal, RT: 102 - 103 MΩ S.S V attenuation (somamtip): 1.1 - 2 S.S attenuation (tipmsoma): 2 - 15 Excitable properties** Ca+2 channels (L, N, P - type) – local dendritic Ca+2 AP Fast activating/inactivating Na+ current – supports backpropagating AP IA and Ih currents – increased density with distance from soma. Synaptology No. synapses/neuron 500 - 200,000 +Type I (excitatory) 60% - 90%: distributed, majority on spines. +Type II (inhibitory) 10% - 40%: near soma, some on spines. #Excitatory synaptic input: non-NMDA: gpeak 0.1 - 0.3 nS; tpeak: 0.3 - 1ms NMDA: gpeak :0.05 - 0.5nS; tpeak: 5 - 50 ms $Inhibitory synaptic input: GABAA: gpeak: 0.4 - 1nS; tpeak: 0.2 - 1.2 ms GABAB: gpeak : 0.1 - 0.3 nS; tpeak: 40 - 150ms It was found that synapses can be identified by type: 1. Type I – Excitatory – Asymmetrical structure so that the post-synaptic membrane is thicker than the pre-synaptic one. 2. Type II – Inhibitory – Symmetrical structure. Cable theory right|thumb The main question was "Are dendrites important for computation or just for connectivity". Eccles claimed they were not important but his student W. Rall said they were important in explaining the experimentally recorded transients and went on to create the "Cable Theory". Ralls' motivation was based on that: • Most of the input current flows into the dendrites • Transients are faster than the membrane time constant • Dendrites are non-isopotential (voltage attenuation, somatic EPSP shape changes with synaptic location) Why Model in Detail? right|400px • Correct interpretation of experimental results (provides predictions!) • Gain insights into key biophysical parameters (enables compact description of the physiological behavior studied capturing the essence e.g., HH model for the AP) • Suggest possible computational (functional) role for the modeled system (e.g., M&P neuron; Rall’s idea on computing direction of motion by single neuron; gain control+ orientation tuning etc. for the cortical column) ימין|400px This graph (Rinzel & Rall, 1974) depicts both a steady state (s.s.) voltage and the AOC of a transient voltage. We can see that an injection of current at the soma or center of a dendrite acts as if the dendrites are simple cylinders (it doesn't mind the split ups) while injecting current at the tips of the dendrites propagates very differently into different regions due to those split ups. Attenograms – Morpho electrotonic transform: A technique to model the structure of the dendrites by their response to input currents rather than by their physical properties. This method is sensitive to the input frequency due to the fact that at high frequencies the capacitance of the membrane filters the current and attenuates it rapidly causing for larger L values (electrotonic lengths) of the dendrites, meaning that it's harder to get to the tip of a dendrite from the soma. ימין|400px Transient condition – infinite cable You can approximate the transient velocity (EPSP peak moving along the dendrite) by 2λ/τ. We get a relatively slow velocity, meaning the dendrite is acting as a delay line. ימין|400px Rall 1967 – The electrical distance (from the soma) of the current injection causes a different time to peak (x axis) and shape (y axis, width at half of the peak voltage). We see that it's a monotonic rising function of the location (in both axis). Compartmental Modeling שמאל|400px ימין|300px The main idea is to model the neuron as many iso-potential compartments, each compartment modeled as an electrical circuit connected to the other compartments. These models are done mostly using programs like NEURON or GENESIS which create the multiple paired ODEs which are then solved numerically for a given input. When the density of the compartments is high enough (Each compartment is smaller than λ/10) then the numerical solution converges to that of the continuous model. • The advantage of the compartmental model is not in the passive case (which is analytically solvable) but in the ability to include active components in the model. • The loss of structural information is already done in the mapping of the dendrites in a discrete manner, so compartmentalizing does not necessarily incurs further loss of information. Nonlinear Dendrites Dendritic Spines – the spines on dendrites can be reconstructed using multiple layers SEM of the dendrites. The modeling of the spines is done in smaller compartments modeling each spine individually. Recording From neurons (in different locations) ממוזער|ימין|100px DIC video microscopy – A method to see unstained neurons (bodies and dendrites). right|300px Back propagating action potential – in 94 Stuart & Sakmann used DIC and coloring to patch clamp the same neuron in its soma and dendrite to record simultaneously from both locations. They found that the spike created at the soma travels backwards into the dendrite. In later works they also recorded from the axon (very hard since it's 1-1.5 microns in diameter) where they saw that the spike (EPSP) comes before the spike in the soma, so the soma actually also gets the spike in backpropagation. They chemically deactivated the active channels in the dendrites to test the dependency of the backpropagation on them. They found that the backpropagation to the dendrites depended very much on those channels and didn't go far without them. They also found that while there are about 1/30 of the active sodium channels in the apical dendrites as in the soma, these channels were enough for the backpropagation. In a later work they found that the backpropagating spike is critical for LTP and LTD signals. synapses in very distal dendrites do not receive enough of the backpropagating spike to act as a signal and they also use the Calcium Spike as a signal. right|300px Larkoum & Sakmann: (b) current injection to the dendrite mimicking local activity. (d) current injection both to the soma and the dendrite. A dendritic spike appears, caused by the Calcium channels. (e) The same dendritic spike effect by a large local current injection at the dendrite. The calcium spike also propagates to the soma and can cause repeated spiking and bursting. The burst can be used as a readout mechanism depending on its existence. Distal synapses still maintain their plasticity even if the active sodium channels are blocked and no backpropagating spiked are present. Sufficient local activity will create a calcium spike which can facilitate hebbian plasticity. Cell type classification Cortical neurons are morphologically diverse. Around 20% of them are inhibitory, mostly inter- neurons, and their firing patterns are very diverse. Work is being done to try and cluster these cells into families of firing patterns and understand the importance of this diversity which is not common to all cells (pyramidal cells are not so diverse). מרכז|600px Another important question is the stability of these firing patterns. Are they stable or do they change dynamically with different states of the cell. Are they emerging from intrinsic properties of the cells (types of channels) or from the network connectivity? Today some of the classification is done by DNA expression in neurons. Computing with Dendrites right|thumb|200px • 1983 - McCulloch & Pits - publish an article on the possible logical calculations done with linear point neurons. • Later – Koch & Pogio – expand the model to dendrites with extensive structures where the locations of the synapses affect the final calculation. The model assumes inhibition only negates the calculations done in more distal locations. • Direction selectivity – It was discovered that there are single ganglion cells in th retina that respond to movement in a certain direction. • 1964 – Rall – Suggests a model where the order of inputs (distal-proximal vs. proximal- distal) determines whether the cell fires or not. This happens since the preferred direction (distal-proximal) causes the inputs to sum as they come since the current propagates in the same direction as the inputs. While in the opposite direction the input at the beginning causes the soma to saturate and not integrate further inputs that in their turn get farther and farther away from the soma. This of course depends on the sweep velocity of the inputs on the dendrite and the integrating time constant. These types of direction selectivity models rely on the connectivity of the inputs in an orderly manner to the dendrites. It doesn't seem that there are evident supporting this condition in the brain and specifically in the retina. left|400px right|400px Open Questions • Noise and Stochastic behavior – The classical models like H&H are deterministic, yet the recordings from cells, even with repeated similar inputs, are variable. The source of this variability is the stochastic operation of the synapses, the noise created by the release of vesicles and their reception. It seems that this noise is not averaged and even the spike generator is noisy as well. This can be modeled by adding probability mechanisms into deterministic models, the question remains what is the importance of this noise and variability. • The role of dendrites in plasticity • Reduction and simplifying – There is no consensus regarding the sufficient similarity of a simplified model to the real deal. The reduction done in modeling can be morphological or electrical (the way you represent the channels) but what is a good measure to say that one simple model is similar to the other more complex one? מרכז|500px